该研究用磁化铁氧体圆盘的圆形超导微波谐振器实验检验 Rosenzweig-Porter 模型，实现从 Poisson 到 Gaussian 酉系综统计的转变，并用约1000个本征频率及 S 矩阵涨落验证理论。
摘要：
我们报告了一项实验研究，考察一个具有可积经典动力学的量子系统向时间反演（T）不变性被破坏且具有混沌经典对应体的系统的转变。高精度实验采用圆形平面超导微波谐振器开展，其中 T 不变性破坏和混沌性通过磁化置于其中心的铁氧体圆盘产生；该圆盘在第一横电模截止频率以上充当随机势。我们测定了约1000个本征频率的完整序列，并发现其与 Rosenzweig-Porter（RP）模型谱性质的解析预测高度吻合；该模型在典型可积系统预期的 Poisson 统计与 T 不变性被破坏的混沌系统所预测的 Gaussian 酉系综统计之间进行插值。此外，我们将 RP 模型与用于量子混沌散射的 Heidelberg 方法相结合，为相应开放量子系统的散射（S）矩阵构建了一个随机矩阵模型，并证明其完美再现了所测微波谐振器 S 矩阵的涨落性质。
英文摘要：
We report on an experimental investigation of the transition of a quantum system with integrable classical dynamics to one with violated time-reversal (T) invariance and chaotic classical counterpart. High-precision experiments are performed with a flat superconducting microwave resonator with circular shape in which T-invariance violation and chaoticity are induced by magnetizing a ferrite disk placed at its center, which above the cutoff frequency of the first transverse-electric mode acts as a random potential. We determine a complete sequence of approx. 1000 eigenfrequencies and find good agreement with analytical predictions for the spectral properties of the Rosenzweig-Porter (RP) model, which interpolates between Poisson statistics expected for typical integrable systems and Gaussian unitary ensemble statistics predicted for chaotic systems with violated T invariance. Furthermore, we combine the RP model and the Heidelberg approach for quantum-chaotic scattering to construct a random-matrix model for the scattering (S) matrix of the corresponding open quantum system and show that it perfectly reproduces the fluctuation properties of the measured S matrix of the microwave resonator.
Experimental study of coupled quantum billiards with integrable and chaotic classical dynamics and test of a special Rosenzweig-Porter model
Lanzhou University（兰州大学）；Institute for Basic Science（韩国基础科学研究院）；Qilu Normal University（齐鲁工业大学）；University of Electronic Science and Technology of China（电子科技大学）；Korea University of Science and Technology (UST)（韩国科学技术院）
The statistical distribution of eigenfunctions for the Rosenzweig-Porter model is derived for the region where eigenfunctions have fractal behaviour. The result is based on simple physical ideas and leads to transparent explicit formulas which agree very well with numerical calculations. It constitutes a rare case where a non-trivial eigenfunction distribution is obtained in a closed form.
The Rosenzweig-Porter model has seen a resurgence in interest as it exhibits a non-ergodic extended phase between the ergodic extended metallic phase and the localized phase. Such a phase is relevant to many physical models from the Sachdev-Ye-Kitaev model in high-energy physics and quantum gravity, to the interacting many-body localization in condensed matter physics and quantum computing. This phase is characterized by fractal behavior of the wavefunctions, and a postulated correlated mini-band structure of the energy spectrum. Here we will seek evidence for the latter in the spectrum. Since…
Spectral properties of Levy Rosenzweig-Porter model via supersymmetric approach
Nanocenter CENN（纳米中心CENN）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Jozef Stefan Institute（约瑟夫·斯特凡研究所）；Abdus Salam International Center for Theoretical Physics（阿卜杜斯·萨拉姆国际理论物理中心）
We consider the static and dynamic phases in a Rosenzweig-Porter (RP) random matrix ensemble with the tailed distribution of off-diagonal matrix elements of the form of the large-deviation ansatz. We present a general theory of survival probability in such a random-matrix model and show that the {\it averaged} survival probability may decay with time as the simple exponent, as the stretch-exponent and as a power-law or slower. Correspondingly, we identify the exponential, the stretch-exponential and the frozen-dynamics phases. As an example, we consider the mapping of the Anderson model on Ran…
Fading ergodicity and quantum dynamics in random matrix ensembles
Georg-August-Universität Göttingen（哥廷根大学）；J. Stefan Institute（约韦·斯特凡研究所）；University of Ljubljana（卢布尔雅那大学）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）
The Gaussian Orthogonal Ensemble (GOE) of random matrices has been widely employed to describe diverse phenomena in strongly coupled quantum systems. An important prediction is that the decay rates of the GOE eigenstates fluctuate according to the distribution for one degree of freedom, as derived by Brink and by Porter and Thomas. However, we find that the coupling to the decay channels can change the effective number of degrees of freedom from $ν= 1$ to $ν= 2$. Our conclusions are based on a configuration-interaction Hamiltonian originally constructed to test the validity of transition-state…
The counterintuitive fact that wave chaos appears in the bending spectrum of free rectangular thin plates is presented. After extensive numerical simulations, varying the ratio between the length of its sides, it is shown that (i) frequency levels belonging to different symmetry classes cross each other and (ii) for levels within the same symmetry sector, only avoided crossings appear. The consequence of anticrossings is studied by calculating the distributions of the ratio of consecutive level spacings for each symmetry class. The resulting ratio distribution disagrees with the expected Poiss…
Matrix models showing chaotic-integrable transition in the spectral statistics are important for understanding Many Body Localization (MBL) in physical systems. One such example is the $β$-ensemble, known for its structural simplicity. However, eigenvector properties of $β$-ensemble remain largely unexplored, despite energy level correlations being thoroughly studied. In this work we numerically study the eigenvector properties of $β$-ensemble and find that the Anderson transition occurs at $γ= 1$ and ergodicity breaks down at $γ= 0$ if we express the repulsion parameter as $β= N^{-γ}$. Thus o…
We study the statistics of the local resolvent and non-ergodic properties of eigenvectors for a generalised Rosenzweig-Porter $N\times N$ random matrix model, undergoing two transitions separated by a delocalised non-ergodic phase. Interpreting the model as the combination of on-site random energies $\{a_i\}$ and a structurally disordered hopping, we found that each eigenstate is delocalised over $N^{2-γ}$ sites close in energy $|a_j-a_i|\leq N^{1-γ}$ in agreement with Kravtsov \emph{et al}, arXiv:1508.01714. Our other main result, obtained combining a recurrence relation for the resolvent mat…
In this paper we study the out-of-equilibrium phase diagram of the quantum version of Derrida's Random Energy Model, which is the simplest model of mean-field spin glasses. We interpret its corresponding quantum dynamics in Fock space as a one-particle problem in very high dimension to which we apply different theoretical methods tailored for high-dimensional lattices: the Forward-Scattering Approximation, a mapping to the Rosenzweig-Porter model, and the cavity method. Our results indicate the existence of two transition lines and three distinct dynamical phases: a completely many-body locali…
We study quantum tomography from a continuous measurement record obtained by measuring expectation values of a set of Hermitian operators obtained from unitary evolution of an initial observable. For this purpose, we consider the application of a random unitary, diagonal in a fixed basis at each time step and quantify the information gain in tomography using Fisher information of the measurement record and the Shannon entropy associated with the eigenvalues of covariance matrix of the estimation. Surprisingly, very high fidelity of reconstruction is obtained using random unitaries diagonal in …
Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2} >= b {\cal F}(|i-j|) \ll 1$. We perform a regular expansion of the spectral form-factor $K(τ) = 1 + b K_{1}(τ) + b^{2} K_{2}(τ) + ... $ in powers of $b \ll 1$ with the coefficients $K_{m}(τ)$ that take into account interaction of (m+1) energy levels. To calculate $K_{m}(τ)$, we develop a diagrammatic technique which is based on the Trotter formula and on th…
